# A triangle has corners A, B, and C located at #(7 ,6 )#, #(4 ,3 )#, and #(5 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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To find the altitude going through corner C, we first need to determine the equation of the line containing side AB, which is the base of the triangle. Then, we'll find the midpoint of side AB, which will be a point on the altitude. After that, we'll find the equation of the perpendicular line passing through C and the midpoint of AB, which will be the altitude. Finally, we'll find the intersection point of this altitude with side AB, which will give us the endpoints of the altitude, and calculate the distance between these endpoints to find the length of the altitude.

Step 1: Find the equation of line AB: [ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) ]

[ y - 6 = \frac{3 - 6}{4 - 7}(x - 7) ]

[ y - 6 = -1(x - 7) ]

[ y - 6 = -x + 7 ]

[ y = -x + 13 ]

Step 2: Find the midpoint of AB: [ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ]

[ \text{Midpoint} = \left(\frac{7 + 4}{2}, \frac{6 + 3}{2}\right) ]

[ \text{Midpoint} = \left(\frac{11}{2}, \frac{9}{2}\right) ]

Step 3: Find the equation of the altitude passing through C and the midpoint of AB: Since the slope of line AB is -1, the slope of the altitude passing through C and the midpoint of AB will be 1 (the negative reciprocal of -1).

[ \text{Equation of altitude}: y - y_1 = m(x - x_1) ]

[ y - 8 = 1(x - 5) ]

[ y - 8 = x - 5 ]

[ y = x + 3 ]

Step 4: Find the intersection point of the altitude with side AB: [ -x + 13 = x + 3 ]

[ -2x = -10 ]

[ x = 5 ]

[ y = 5 + 3 = 8 ]

The intersection point is (5, 8), which is also the point C. Therefore, the altitude goes through point C.

Step 5: Calculate the distance between point C and the midpoint of AB: [ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

[ \text{Distance} = \sqrt{(5 - 5)^2 + (8 - 4.5)^2} ]

[ \text{Distance} = \sqrt{0 + (3.5)^2} ]

[ \text{Distance} = \sqrt{12.25} ]

[ \text{Distance} = 3.5 ]

Therefore, the length of the altitude going through corner C is 3.5 units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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